On the Chromatic Index of Random Uniform Hypergraphs

Abstract

Let $\mathbb{H}^{(k)}(n, N)$, where $k \ge 2$, be a random hypergraph on the vertex set $[n] = \{1, 2, \dots, n\}$ with $N$ edges drawn independently with replacement from all subsets of $[n]$ of size $k$. For $\bar{d} = k N/n$ and any $\varepsilon > 0$ we show that if $k = o(\ln ({\bar d}/\ln n))$ and $k = o(\ln (n/\ln {\bar d}))$, then with probability $1-o(1)$ a random greedy algorithm produces a proper edge coloring of $\mathbb{H}^{(k)}(n, N)$ with at most $\bar{d} (1+\varepsilon)$ colors. This yields the asymptotic chromatic number of the corresponding uniform random intersection graph.